We study the long-time evolution of the trailing shelves that form behind solitary waves
moving through an inhomogeneous media, within the framework of the variable-coeffecient
Korteweg-de Vries equation. We show that the nonlinear evolution of the shelf leads typically
to the generation of an undular bore and an expansion fan, which form apart but start to
overlap and nonlinearly interact after a certain time interval. The interaction zone expands
with time and asymptotically as time goes to infinity occupies the whole perturbed region.
Its oscillatory structure strongly depends on the sign of the inhomogeneity gradient of the
variable background medium. We describe the nonlinear evolution of the shelves in terms
of exact solutions to the KdV-Whitham equations with natural boundary conditions for
the Riemann invariants. These analytic solutions, in particular, describe the generation of
small 'secondary' solitary waves in the trailing shelves, a process observed earlier in various
numerical simulations.

Description:

This pre-print has been submitted, and accepted, to the journal, Chaos. The definitive version: EL, G.A. and GRIMSHAW, R.H.J., 2002. Generation of undular bores in the shelves of slowly-varying solitary waves. Chaos, 12(4), pp. 1015-1026, is available at: http://chaos.aip.org/chaos/.